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This paper deals with the existence of <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-periodic solutions for <i>n</i>th-order ordinary differential equation involving fixed delay in Banach space <i>E</i>. <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mi>n</mi></msub><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>=</mo><mi>f</mi><mrow><mo>(</mo><mi>t</mi><mo>,</mo><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>,</mo><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>−</mo><mi>τ</mi><mo>)</mo></mrow><mo>)</mo></mrow><mo>,</mo><mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace><mspace width="3.33333pt"></mspace><mi>t</mi><mo>∈</mo><mi mathvariant="double-struck">R</mi><mo>,</mo></mrow></semantics></math></inline-formula> where <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>L</mi><mi>n</mi></msub><mi>u</mi><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>:</mo><mo>=</mo><msup><mi>u</mi><mrow><mo>(</mo><mi>n</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow><mo>+</mo><mstyle displaystyle="true"><munderover><mo>∑</mo><mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow><mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></munderover></mstyle><msub><mi>a</mi><mi>i</mi></msub><msup><mi>u</mi><mrow><mo>(</mo><mi>i</mi><mo>)</mo></mrow></msup><mrow><mo>(</mo><mi>t</mi><mo>)</mo></mrow></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><msub><mi>a</mi><mi>i</mi></msub><mo>∈</mo><mi mathvariant="double-struck">R</mi></mrow></semantics></math></inline-formula>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>i</mi><mo>=</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>,</mo><mo>⋯</mo><mo>,</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow></semantics></math></inline-formula>, are constants, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>f</mi><mo>(</mo><mi>t</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>:</mo><mi mathvariant="double-struck">R</mi><mo>×</mo><mi>E</mi><mo>×</mo><mi>E</mi><mo>⟶</mo><mi>E</mi></mrow></semantics></math></inline-formula> is continuous and <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mi>ω</mi></semantics></math></inline-formula>-periodic with respect to <i>t</i>, <inline-formula><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"><semantics><mrow><mi>τ</mi><mo>></mo><mn>0</mn></mrow></semantics></math></inline-formula>. By applying the approach of upper and lower solutions and the monotone iterative technique, some existence and uniqueness theorems are proved under essential conditions.